Finite Group Databasebeta

← #29779782

Subgroups — #29779782

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451798C211yesmaximal, characteristic, complemented
2#21448526C211yesmaximal, characteristic
3#21447385C211yesmaximal, characteristic, complemented
4#21447380C211yesmaximal, characteristic, complemented
5#21447057C211yesmaximal, characteristic
6#21447057C211yesmaximal, characteristic
7#21446514C211yesmaximal, characteristic
8#21446506C211yesmaximal, characteristic, complemented
9#21446500C211yesmaximal, characteristic
10#21446492C211yesmaximal, characteristic, complemented
11#21445826C211yesmaximal, characteristic
12#21445824C211yesmaximal, characteristic, complemented
13#21445820C211yesmaximal, characteristic
14#21445678C211yesmaximal, characteristic, complemented
15C61×#1805C211yesmaximal, characteristic, complemented
16C2×#14519766C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839704—21no—
28#16839704—21no—
29#16839694—21no—
30#16839694—21no—
31#16839669C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839653—21no—
38#16839653—21no—
39#16839633—21no—
40#16839633—21no—
41#16839632C2211yescharacteristic
42#16839631C2211yescharacteristic
43#16839608C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519766—22no—
64#14519766—22no—
65C4×D244—22no—
66C4×D244—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C4⋊#92418—22no—
86#14519701—22no—
87#14519701—22no—
88#14519699—22no—
89#14519699—22no—
90#14519697—41no—
91#14519697—41no—
92#14519687—41no—
93C8×Dic61—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C4×Q8×C61—22no—
108C4×Q8×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—22no—
116#103×C61—41no—
117#103×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244#4112yes—
147C2×C4×D61—42no—
148C2×C4×D61—81no—
149C2×C4×D61—81no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C42×C61—42no—
164C42×C61#4112yes—
165C42×C61#4112yes—
166#11758198—24no—
167#11758198—24no—
168#11758198—42no—
169#11758198—81no—
170#11758198—81no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—81no—
178C4⋊Dic61—81no—
179C4⋊Dic61—81no—
180C4⋊Dic61—81no—
181#11758196—81no—
182#11758196—81no—
183#11758196—81no—
184#11758196—81no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61#12412yes—
224C2×C4×C61#12412yes—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—84no—
257C8×C61—41no—
258C8×C61—41no—
259#92418—41no—
260#92418—41no—
261#29134—611nomaximal, Sylow
262C22×C61#52511yescharacteristic
263C22×C61C2×#12411yescharacteristic
264C22×C61#53711yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57811yescharacteristic
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#55211yescharacteristic
283C4×C61#55211yescharacteristic
284C4×C61#55311yescharacteristic
285C4×C61#55311yescharacteristic
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293#2647—611no—
294#2326—611no—
295#2100—611no—
296#2099—611no—
297#2042—611no—
298#2042—611no—
299#1936—611no—
300#1935—611no—
301#1934—611no—
302#1933—611no—
303#1805D6111yescharacteristic, complemented
304#1804—611no—
305#1803—611no—
306#1802—611no—
307#1768—611no—
308C2×C61#256211yescharacteristic
309C2×C61#311911yescharacteristic
310C2×C61#312011yescharacteristic
311C2×C61#313611yescharacteristic
312C2×C61#313711yescharacteristic
313C2×C61#326211yescharacteristic
314C2×C61#326311yescharacteristic
315D61—82no—
316C2×#124—611no—
317C2×C4×Q8C2×D6111yescharacteristic
318C2×C4×D4—611no—
319C2×C4⋊C8—611no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#103—611no—
322C2×#103—611no—
323C2×#103C2×D6111yescharacteristic
324C2×#103C2×D6111yescharacteristic
325C2×#102—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#97—611no—
330C2×C4×C8—611no—
331#400—1221no—
332#400—1221no—
333#398—1221no—
334#398—1221no—
335#391—611no—
336#390—611no—
337#389—611no—
338#389—611no—
339#389—611no—
340#389—611no—
341#388—1221no—
342#388—1221no—
343#385—1221no—
344#385—1221no—
345#384—611no—
346#384—611no—
347#382C2×D6111yescharacteristic
348#376—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340C2×D6111yescharacteristic
356#340C2×D6111yescharacteristic
357#336—611no—
358#336—611no—
359C61#2913411yescharacteristic, complemented, Sylow
360C22×Q8C22×D6111yescharacteristic
361C22×D4—611no—
362C23×C4—1221no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367#124—1222no—
368#124—1222no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×D4—1222no—
372C4×D4—1222no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42C22×D6111yescharacteristic
397C4⋊C8—22no—
398C4⋊C8—1222no—
399#103—41no—
400#103—41no—
401#103—1222no—
402#103—1222no—
403#102—1222no—
404#102—1222no—
405#102—2441no—
406#102—2441no—
407#97—2441no—
408C4×C8—2441no—
409#95—21no—
410#95—21no—
411#95—611no—
412#95—611no—
413#95—611no—
414#95—611no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421C24—1221no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1175822612yes—
428C2×D4—612no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—4881no—
446C22×C4—4881no—
447C22×C4—2442no—
448C22×C4D4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4C23×D6111yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—4881no—
477C4⋊C4—4881no—
478C4⋊C4—4881no—
479C4⋊C4—4881no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—4881no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—4881no—
487C22⋊C4—4881no—
488C22⋊C4—2442no—
489C42—42no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—1221no—
494C42#1175822212yes—
495C42#1175822412yes—
496C23—1222no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23C2×D4×D6111yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—1222no—
509D4—1222no—
510D4—2441no—
511D4—2441no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—4884no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4#1451976312yes—
551C2×C4#1451976312yes—
552C2×C4#41×D6111yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—41no—
559C8—41no—
560C8—2441no—
561C8—2441no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22#1684027511yescharacteristic
570C22C2×#1451976311yescharacteristic
571C22#1684034111yescharacteristic
572C22#1684056411yescharacteristic
573C22#1684056911yescharacteristic
574C22#1684057111yescharacteristic
575C22#1684057811yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684045111yescharacteristic
590C4#1684045411yescharacteristic
591C4#1684045911yescharacteristic
592C4#1684046011yescharacteristic
593C2—4882no—
594C2#2145098411yescharacteristic
595C2#2145602511yescharacteristic
596C2#2145604011yescharacteristic
597C2#2145617711yescharacteristic
598C2#2145618011yescharacteristic
599C2#2145727811yescharacteristic
600C2#2145728711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1805D61
hypocenterC61#29134
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21451798C2semidirect
3. #21447385C2semidirect
4. #21447380C2semidirect
8. #21446506C2semidirect
10. #21446492C2semidirect
12. #21445824C2semidirect
14. #21445678C2semidirect
15. C61×#1805C2semidirect
303. #1805D61semidirect
359. C61#29134semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214517982
#214485262
#214473852
#214473802
#214470572
#214465142
#214465062
#214465002
#214464922
#214458262
#214458242
#214458202
#214456782
C61×#18052
#2913461